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Achievability Bounds of Coding with Finite Blocklength for Gaussian Broadcast Channels

Ayşe Ünsal, Jean-Marie Gorce

TL;DR

This work addresses finite-blocklength performance of dirty paper coding on the two-receiver Gaussian broadcast channel. It develops two achievability frameworks by adapting Polyanskiy’s dependence testing bound and the κβ bound to the broadcast setting, introducing per-error-event dispersion analyses and threshold decoding. The authors derive detailed expressions for dispersion-driven error probabilities in both users, including user-specific mis-detection and confusion terms, and characterize the β- and κβ-parameters under DPC with peak-power constraints. The results indicate that dirty paper coding can outperform superposition coding in non-asymptotic regimes, offering guidance for designing finite-blocklength BC systems in practical settings.

Abstract

In this paper, we study the achievable performance of dirty paper coding for the Gaussian broadcast channel (BC) with finite blocklength and we propose two different achievability bounds for this problem. We present the broadcast adaptation of dependence testing bound of Polyanskiy et al. 2010, which is an upper bound on the average error probability that depends on the channel dispersion terms of each error event for fixed input. Additionally, we introduce the $κβ$ lower bounds on the maximal code sizes of each user using dirty paper coding.

Achievability Bounds of Coding with Finite Blocklength for Gaussian Broadcast Channels

TL;DR

This work addresses finite-blocklength performance of dirty paper coding on the two-receiver Gaussian broadcast channel. It develops two achievability frameworks by adapting Polyanskiy’s dependence testing bound and the κβ bound to the broadcast setting, introducing per-error-event dispersion analyses and threshold decoding. The authors derive detailed expressions for dispersion-driven error probabilities in both users, including user-specific mis-detection and confusion terms, and characterize the β- and κβ-parameters under DPC with peak-power constraints. The results indicate that dirty paper coding can outperform superposition coding in non-asymptotic regimes, offering guidance for designing finite-blocklength BC systems in practical settings.

Abstract

In this paper, we study the achievable performance of dirty paper coding for the Gaussian broadcast channel (BC) with finite blocklength and we propose two different achievability bounds for this problem. We present the broadcast adaptation of dependence testing bound of Polyanskiy et al. 2010, which is an upper bound on the average error probability that depends on the channel dispersion terms of each error event for fixed input. Additionally, we introduce the lower bounds on the maximal code sizes of each user using dirty paper coding.
Paper Structure (14 sections, 3 theorems, 48 equations, 3 figures)

This paper contains 14 sections, 3 theorems, 48 equations, 3 figures.

Table of Contents

  1. Introduction
  2. System Model
  3. Error probability
  4. Random codebooks
  5. Related Work
  6. Asymptotic Capacity
  7. Non-asmyptotic performance of SPC
  8. Dependence Testing Bound --Writing on Dirty Paper
  9. Decoding message 1
  10. Decoding message 2
  11. $\kappa \beta$ Bound --Writing on Dirty Paper
  12. Peak power
  13. Derivation of $\beta$
  14. Discussion and Conclusion

Key Result

Theorem 1

An $(n,M_1,M_2,\epsilon,P)$ code exists for the channel as described in Definition eq:def_code_GBC with the average error probability satisfying $\epsilon \leq \epsilon_{SP}$, where $\epsilon_{SP}$ is for $\gamma_j=\log \eta_j$, $\eta_j = (M_j-1)/2$, $j=1,2$ and $\bar{Y}^n_j$ denotes a signal following the same distribution $Y^n_j$ that is independent of the input. $F$ denotes the set of permissi

Theorems & Definitions (6)

  • Definition 2.1
  • Theorem 1: Ünsal and Gorce 2017 UG_2017
  • Theorem 2
  • proof
  • Theorem 3
  • proof